On the structure of the Selberg class, VII: $1\lt d\lt 2$

Abstract

The Selberg class $\mathcal{S}$ is a rather general class of Dirichlet series with functional equation and Euler product and can be regarded as an axiomatic model for the global $L$-functions arising from number theory and automorphic representations. One of the main problems of the Selberg class theory is to classify the elements of $\mathcal{S}$. Such a classification is based on a real-valued invariant $d$ called degree, and the degree conjecture asserts that $d\in\mathbb{N}$ for every $L$-function in $\mathcal{S}$. The degree conjecture has been proved for $d\lt 5/3$, and in this paper we extend its validity to $d\lt 2$. The proof requires several new ingredients, in particular a rather precise description of the properties of certain nonlinear twists associated with the $L$-functions in $\mathcal{S}$.